Reliability Concepts

Detailed discussions of concepts and ideas presented below are available in (3) and

(17). Consider a component with a lifetime, denoted byT, measured in some unit of

‘time’. Usually, the lifetime will be measured in literal time, but it need not always

be the case. Such a T is a non-negative random variable. The reliability function of

this component is defined via

R(t) = 1−F(t) =P r{T > t},

where F(·) is the corresponding distribution function. We assume that lifetime vari- ables are continuous. ForT, its probability density function (pdf) is

f(t) = dF(t)

dt =− dR(t)

dt .

Its hazard rate functionλ(t) is defined as

λ(t)≡lim dt↓0 1 dtP r{t≤T < t+dt|T ≥t}= f(t) R(t),

which can be interpreted as the rate of failure of the component at timet, given that

cumulative hazard function is

Λ(t) =

Z t

0

λ(v)dv.

For a continuous lifetime T, we have the following relationships:

f(t) = λ(t) exp{−Λ(t)} and R(t) = exp{−Λ(t)} (4.1) For a component with lifetimeT, its associated state process is{X(t) :t≥0}, where

X(t) = I{T > t} is a binary variable taking values of 1 or 0 depending on whether the component is still working (1) or failed (0) at time t. The function I(·) denotes indicator function.

Consider a system composed of K components, where this system is either in a

working (1) or failed (0) state. The functionality of a system is characterized by its

structure function

φ:{0,1}K _{→ {0,1},}

withφ(x1, x2,· · · , xK) denoting the state of the system when the states of the compo- nents are x= (x1, x2,· · · , xK)∈ {0,1}K. The vector xis called the component state vector. Such a system is said to be coherent if each component is relevant and the

structure function φ is nondecreasing in each argument. The ith component is rele-

vant if there exists a state vector x∈ {0,1}K _{such that φ(x,0}

i) = 0< 1 = φ(x,1i), with the notation that (x, ai) = (x1, . . . , xi−1, ai, xi+1, . . . , xn). We will only consider

coherent systems in this chapter. Four simple examples of coherent systems are the

(i) series; (ii) parallel; (iii) three-component series-parallel; and (iv) five-component

bridge systems, whose respective structure functions are given by

φser(x1, . . . , xK) = QKi=1xi; (4.2)

φpar(x1, . . . , xK) = `Ki=1xi ≡1−QKi=1(1−xi); (4.3)

φserpar(x1, x2, x3) = x1(x2∨x3); (4.4)

The binary operator ‘∨’ means taking the maximum, i.e. a1 ∨a2 = max(a1, a2) = 1−(1−a1)(1−a2) for ai ∈ {0,1}, i= 1,2.

Let Xi, i= 1, . . . , K, be the state (at a given point in time) random variables for

the K components, and assume that they are independent. Denote bypi = Pr{Xi = 1}, i= 1, . . . , K, and let p = (p1, p2, . . . , pK) ∈[0,1]K be the components reliability vector (at a given point in time). Associated with the coherent structure function φ

is the reliability function defined via

hφ(p) = E[φ(X)] = Pr{φ(X) = 1}.

This reliability function provides the probability that the system is functioning, at

the given point in time, when the component reliabilities at this time arepi’s. For the

first three concrete systems given above, these reliability functions are, respectively:

hser(p1, . . . , pK) =QKi=1pi; (4.6)

hpar(p1, . . . , pK) = `Ki=1pi ≡1−QKi=1(1−pi); (4.7)

hserpar(p1, p2, p3) = p1[1−(1−p2)(1−p3)]; (4.8) For the bridge structure, its reliability function at a given point in time, obtained

first by simplifying the structure function, is given by

hbr(p1, p2, p3, p4, p5) = (p1p4+p2p5+p2p3p4+p1p3p5+ 2p1p2p3p4p5)

−(p1p2p3p4+p2p3p4p5+p1p3p4p5+p1p2p3p5+p1p2p4p5). (4.9) Of more interest, however, is viewing the system reliability function as a function of

time t. Denoting by S the lifetime of the system, we are interested in the function

RS(t) = Pr{S > t}

which is the probability that the system does not fail in [0, t]. Let T= (T1,· · · , TK) be the vector of lifetimes of the K components. The vector of component state

processes is{X(t) = (X1(t)· · · , XK(t)) :t≥0}. The system lifetime is then

S = sup{t≥0 :φ[X1(t),· · · , XK(t)] = 1}.

The component reliability functions are Ri(t) =E[Xi(t)] = Pr{Ti > t}, i= 1, . . . , K. If the component lifetimes are independent, then the system reliability function be-

comes

RS(t) =E[φ(X1(t), . . . , XK(t))] = hφ(R1(t), . . . , RK(t)). (4.10)

That is, under independent component lifetimes, to obtain the system reliability

function, we simply replace thepi’s in the reliability functionhφ(p1, . . . , pK) byRi(t)’s.

For the concrete examples of coherent systems given in (4.2–4.5), we therefore obtain:

Rser(t) =QKi=1Ri(t); (4.11)

Rpar(t) = 1−Qi=1K (1−Ri(t)); (4.12)

Rserpar(t) =R1(t)[1−(1−R2(t))(1−R3(t))]. (4.13) For the bridge structure, in (4.9), we replace each pi by Ri(t) to obtain its system

reliability function. As an illustration of these system reliability functions for the

four examples of coherent systems, with Ri(t) = exp(−λt), i= 1, . . . , K, the system reliability functions are

Rser(t;λ, K) = exp(−Kλt);

Rpar(t;λ, K) = 1−[1−exp(−λt)]K;

Rserpar(t;λ) = exp(−λt)[1−(1−exp(−λt))2];

Rbr(t;λ) = 2 exp{−2λt}+ 2 exp{−3λt}+ 2 exp{−5λt} −5 exp{−4λt}.

When λ = 1 and K = 5, these system reliability functions are plotted in Figure 4.1.

An important concept in reliability is measuring the relative importance of each of

0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 Time System Reliability ● 5−Series 5−Parallel 3−Series−Parallel 5−Bridge

Figure 4.1 System reliability functions for the series, parallel, series-parallel, and bridge systems when the components have common unit exponential lifetimes and there are 5 components in the series and parallel systems.

importance (cf., (3)). We focus on the so-called reliability importance measure as

this will play an important role in the improved estimation of the system reliability.

The reliability importance of component j in aK-component system with reliability

function hφ(·) is

Iφ(j;p) =

∂hφ(p1,· · · , pj−1, pj, pj+1,· · · , pK)

∂pj

=hφ(p,1j)−hφ(p,0j). (4.14) This measures how much system reliability changes when the reliability of component

j changes, with the reliabilities of the other components remaining the same. For a

coherent system, the reliability importance of a component is positive. As examples,

the reliability importance of the jth component in a series system is

Iser(j;p) =

hser(p)

pj

, j = 1, . . . , K,

showing that in a series system the weakest (least reliable) component is the most

reliability importance of thejth component is

Ipar(j;p) =

1−hpar(p) 1−pj

, j = 1, . . . , K,

indicating that the most reliable component is the most important component in a

parallel system. For the 3-component series-parallel system, the reliability importance

of the three components are

Iserpar(1;p) = 1−(1−p2)(1−p3);

Iserpar(2;p) =p1(1−p3);

Iserpar(3;p) =p1(1−p2).

Evaluated at p = (p, p, p), they become Iserpar(1;p) = p(2−p) and Iserpar(2;p) =

Iserpar(3;p) =p(1−p), which confirms the intuitive result that when the components are equally reliable, the component in series (component 1) is the most important

component. In general, however, component 1 is not always the most important.

For instance, if components 2 and 3 are equally reliable with reliability p2, then the

reliability importance of components 1, 2 and 3 become

Iserpar(1; (p1, p2, p2)) =p2(2−p2);

Iserpar(2; (p1, p2, p2)) =Iserpar(3; (p1, p2, p2)) = p1(1−p2).

In this case, component 2 (and 3) is more important than component 1 whenever

p1(1−p2)> p2(2−p2), or equivalently, p1 > p2(2−p2) 1−p2 .